Looking for proof of Superpositional Energy Conservation

[Dale]

If we really need to abandon infinitesimally small slit widths

It is an approximation. It is never "true", but it can be "valid" (meaning "close enough"). Its validity depends entirely on what you want to use it for, and when you get a clearly wrong answer it is always good to go back and check your approximations. For the purpose of post 26, the approximation gives an answer that is no longer "close enough", so it is not a valid approximation.

For a slit-spacing and wavelength ratio of 1, the superposed area is 22% larger than the un-superposed one. That's a pretty large discrepancy when we're trying to make the case that there should be no discrepancy at all.

What are you referring to here? What is this superposed area and what is the case are you trying to make?

 

[greswd]

What are you referring to here? What is this superposed area and what is the case are you trying to make?

The area under the intensity function that I first mentioned in #18.

You can play with the Desmos widget too:
https://www.desmos.com/calculator/8gcwydxb4t

 

[Dale]

The area under the intensity function that I first mentioned in #18

Again neither the intensity nor the Poynting vector obey superposition. As far as I understand what you are doing, the claim that there is no discrepancy is false, and it is not a case that I am trying to make.

The fields obey Maxwell's equations, which is linear and therefore the fields obey superposition. The fields also conserve energy, per Poynting's theorem. The intensity is the magnitude of the Poynting vector, so it is only a part of the energy conservation equation. Furthermore, the Poynting vector itself does not obey superposition.

Superposition and energy conservation are separate concepts.

 

[greswd]

Again neither the intensity nor the Poynting vector obey superposition. The claim that there is no discrepancy is false, and it is not a case that I am trying to make.

The fields obey Maxwell's equations, which is linear and therefore the fields obey superposition. The fields also conserve energy, per Poynting's theorem. The intensity is the magnitude of the Poynting vector, so it is only a part of the energy conservation equation. Furthermore, the Poynting vector itself does not obey superposition.

Superposition and energy conservation are separate concepts.

If the intensity is only a part of the energy conservation equation, what's the other part?

 

[Dale]

If the intensity is only a part of the energy conservation equation, what's the other part?

See the first equation here:

https://en.m.wikipedia.org/wiki/Poynting's_theorem

The intensity is just the magnitude of S.